Singular Value Consulting

Variations on factorial!

If you’ve heard of factorial, have you heard of double factorial or subfactorial?

Double factorial is written n!!. The factorial of a positive integer n is the product of all positive integers less than or equal to n. The double factorial of n is the product of all integers less than or equal to n that have the same parity. That is, for an odd number n, the product defining n!! includes only odd integers and for an even integer n, the product defining n!! includes only even integers. For example, 7!! = 7 × 5 × 3 × 1 and 8!! = 8 × 6 × 4 × 2. By definition, 0!! and -1!! equal 1.

Double factorials often arise in integrals and power series and make it possible to state equations succinctly that would be verbose otherwise. For example,

It’s possible to define higher factorials or multifactorials. For instance n!!!, the triple factorial of n, is the product of positive integers less than n and congruent to n mod 3. So, for example, 8!!! = 8 × 5 × 2.

Factorials count the number of ways a set can be arranged. A set with n distinct elements can be arranged in n! ways. The number of arrangements that move every element from its original position is the subfactorial of n. Sometimes subfactorial is written with the exclamation point in front of its argument and sometimes it is written with an inverted exclamation point following its argument, i.e.

(By the way, the inverted exclamation mark, used to mark the beginning of an exclamatory sentence in Spanish, is Unicode character U+00A1. You can produce it in HTML with &iexl;. In TeX, you can produce it !` outside of math mode and mbox{!`} in math mode.)

Subfactorial can be computed from the factorial by

for positive n where ⌊x⌋ is the greatest integer less than x. The subfactorial of 0 is defined to be 1.

I have been thinking about a generalization of factorials along the lines of operation. So the normal factorial would be the multiplicative factorial. The additive factorial of n would be sum of the first n integers. The exponential factorial would be n^n-1^n-2^….. It’s a neat way to turn any binary opp on the naturals into a monary opp. Any ideas what these are called?